Fix \(j=2,\dots ,m\)  and \(\hat{x}_j \in \mathbb {R}^{m-2}\) .
Let \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\)  and let \(\epsilon >0\) . In the similar way in [1]} and in [2]}, Lemma 5.1 we consider the functions \(\xi _1(x,y_1), \dots , \xi _n(x,y_n)\)  and \(h(x)\)  defined for small \(\delta >0\)  by
\(\xi _s (x,y_s)= \left\lbrace \begin{array}{l}\quad \quad 0\hspace{91.04872pt} -T<x<T, \, \, -r_s <y_s\le \gamma _s(x) -\epsilon -\delta \\\frac{1}{\delta }(y_s -\gamma _s(x)+\epsilon +\delta ) \hspace{18.49411pt} -T<x<T, \, \, \gamma _s (x) -\epsilon -\delta <y_s\le \gamma _s (x) -\epsilon \\\quad \quad 1 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) -\epsilon <y_s\le \gamma _s(x)\\\frac{1}{\delta }(-y_s +\gamma _s(x) +\delta ) \hspace{28.45274pt} -T<x<T, \, \, \gamma _s (x) <y_s\le \gamma _s(x) +\delta \\\quad \quad 0 \hspace{91.04872pt} -T<x<T, \, \, \gamma _s(x) +\delta <y_s< r_s\end{array}\right.\) 
for \(s=1,\dots ,n\)  and
\(h(x) = \left\lbrace \begin{array}{l}\quad \quad 0\hspace{71.13188pt} -T< x \le t_1 -\delta \\\frac{1}{\delta }(x-t_1 +\delta ) \hspace{36.98866pt} t_1 -\delta <x\le t_1\\\quad \quad 1 \hspace{72.55461pt} t_1 <x\le t_2\\\frac{1}{\delta }(-x+t_2 +\delta ) \hspace{29.87547pt} t_2<x \le t_2+\delta \\\quad \quad 0 \hspace{73.97733pt}t_2+\delta <x<T\end{array}\right.\) 
If \(\phi \)  is distributional solution of \((D_2^\phi \phi , \dots , D_m^\phi \phi )=w\)  in \(\mathcal {O} \) , for all test function \(\varphi \)  we have
\(\begin{aligned}\int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi (x,\hat{x}_j,y)\varphi _x(x,y) & +f_1(\phi (x,\hat{x}_j,y)) \varphi _{y_1}(x,y)+\dots +f_n(\phi (x,y)) \varphi _{y_n}(x,y)\\& +w_j(x,\hat{x}_j,y)\varphi (x,y))\, dx dy =0\end{aligned}\) 
where
\(\begin{aligned}f_s(\phi (x_j,\hat{x}_j,y))& = \frac{1}{2} \biggl ( b_{j1}^{(s)}\phi ^2(x_j,\hat{x}_j,y)+\phi (x_j,\hat{x}_j,y)\sum _{l=2}^{m}b_{jl}^{(s)}(\hat{x}_j)_l \biggl )\end{aligned}\) 
for all \((x_j,\hat{x}_j,y)\in \mathcal {O} \)  and \(s=1,\dots ,n\) .
We choose the test function \(\varphi (x,y_1,\dots ,y_n )= \xi _1 (x,y_1) \dots \xi _n (x,y_n) h(x)\)  and compute the limit \(\delta \rightarrow 0^+\) :
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \dots \int _{-r_n}^{r_n} (\phi \varphi _x+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 \dots \xi _n h+\dots +f_n(\phi ) \xi _1 \dots (\xi _n)_{y_n}h+w_j\xi _1 \dots \xi _nh)\, dx dy\end{aligned}\) 
For simplicity we consider the case \(n=2\) . We compute the following limit:
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} & \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \biggl (\phi \biggl ((\xi _1)_x\xi _2 h+\xi _1(\xi _2)_x h+\xi _1\xi _2 h_x\biggl )+f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h \\& \quad +f_2(\phi ) \xi _1(\xi _2)_{y_2}h + w_j\xi _1 \xi _2h\biggl )\, dx dy_1dy_2=: \lim _{\delta \rightarrow 0^+} I_1+I_2+I_3+I_4+I_5+I_6.\end{aligned}\) 
Then
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_1& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi (\xi _1)_x\xi _2 h\, dx dy_1dy_2\\& =\int _a^b \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_2& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1(\xi _2)_x h\, dx dy_1dy_2\\& =\int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\\\end{aligned}\) 
\(\begin{aligned}\lim _{\delta \rightarrow 0^+} I_3& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} \phi \xi _1\xi _2 h_x\, dx dy_1dy_2\\& = \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2 -\int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\\lim _{\delta \rightarrow 0^+} I_4& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_1(\phi ) ( \xi _1)_{y_1} \xi _2 h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \Bigl ( f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2)) \Bigl )\, dxdy_2\\\lim _{\delta \rightarrow 0^+} I_5& = \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2} f_2(\phi )\xi _1 ( \xi _2)_{y_2} h\, dx dy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \Bigl ( f_2(\phi (x,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,y_1, \gamma _2 (x) )) \Bigl )\, dxdy_1\\\lim _{\delta \rightarrow 0^+} I_6 &= \lim _{\delta \rightarrow 0^+} \int _{-T}^T \int _{-r_1}^{r_1} \int _{-r_2}^{r_2}w_j\xi _1 \xi _2h\, dx dy_1dy_2 = \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j\, dxdy_1dy_2\end{aligned}\) 
Hence we obtain
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} f_1(\phi (x,\hat{x}_j,\gamma _1 (x)- \epsilon ,y_2))-f_1(\phi (x,\hat{x}_j,\gamma _1 (x) ,y_2))\\& \quad + \dot{\gamma }_1(x) \Bigl (\phi (x,\hat{x}_j,\gamma _1(x), y_2) - \phi (x,\hat{x}_j,\gamma _1 (x)-\epsilon , y_2)\Bigl )\, dx dy_2 \\& \quad + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x)- \epsilon ))-f_2(\phi (x,\hat{x}_j,y_1, \gamma _2 (x) )) \\&\quad + \dot{\gamma }_2(x) \Bigl (\phi (x,\hat{x}_j,y_1, \gamma _2(x)) - \phi (x,\hat{x}_j,y_1, \gamma _2 (x)-\epsilon )\Bigl )\, dxdy_1\end{aligned}\) 
From (REF ) with \(n=2\)  we deduce that
\(\begin{aligned}& \int _{\gamma _1(t_2)-\epsilon }^{\gamma _1(t_2)} \int _{\gamma _2(t_2)-\epsilon }^{\gamma _2(t_2)} \phi (t_2,\hat{x}_j,y_1, y_2) \, dy_1dy_2 - \int _{\gamma _1(t_1)-\epsilon }^{\gamma _1(t_1)} \int _{\gamma _2(t_1)-\epsilon }^{\gamma _2(t_1)} \phi (t_1,\hat{x}_j,y_1, y_2) \, dy_1dy_2\\& \quad - \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)}\int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} w_j (x,\hat{x}_j,y_1, y_2) \, dxdy_1dy_2\\& = \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),y_2)+\\& + \phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 + \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )\\ & - \phi (x,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,y_1,\gamma _2(x)) + \phi (x,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\\end{aligned}\) 
If \(b_{j1}^{(1)}, b_{j1}^{(2)} = 0\)  the integral (REF ) is zero. On the other hand if \(b_{j1}^{(1)} , b_{j1}^{(2)}\ne 0\) , because \(\phi \)  is locally \(1/2 \) -Hölder continuous along the vertical components with Hölder's constant \(C_h>0\)
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2)- \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)| }{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,\gamma _1(x),y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_2\in (\gamma _2(x)-\epsilon , \gamma _2(x))\) . Hence dividing by \(\epsilon ^2\)  in \((\ref {lipottobre})\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _2(x)-\epsilon }^{\gamma _2(x)} \frac{1}{2}b_{j1}^{(1)}\biggl ( \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2)- \phi (x,\hat{x}_j,\gamma _1(x) ,y_2) \biggl ) \biggl ( \phi (x,\hat{x}_j,\gamma _1(x),y_2)+\\& \quad + \phi (x,\hat{x}_j,\gamma _1(x)-\epsilon ,y_2) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_2 \\& \le |b_{j1}^{(1)}| \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1).\end{aligned}\) 
Moreover
\(\begin{aligned}&\lim _{\epsilon \rightarrow 0 } \frac{1}{\epsilon ^2} \int _{t_1}^{t_2} \int _{\gamma _1(x)-\epsilon }^{\gamma _1(x)} \frac{1}{2}b_{j1}^{(2)}\biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x) ) \biggl ) \biggl ( \phi (x,\hat{x}_j,y_1,\gamma _2(x))+\\& \quad + \phi (x,\hat{x}_j,y_1, \gamma _2(x)-\epsilon ) -2\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) \biggl ) \, dxdy_1\\& \le |b_{j1}^{(2)} | \frac{(1+\sqrt{2})C_h^2}{2} (t_2-t_1),\end{aligned}\) 
where in the last inequality we have used the fact
\(\begin{aligned}&\frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon )- \phi (x,\hat{x}_j,y_1,\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{|\phi (x,\hat{x}_j,y_1,\gamma _2(x)) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{\epsilon }} \le C_h\\& \frac{ |\phi (x,\hat{x}_j,y_1,\gamma _2(x)-\epsilon ) -\phi (x,\hat{x}_j,\gamma _1(x),\gamma _2(x)) |}{\sqrt{2\epsilon }} \le C_h\end{aligned}\) 
for \(x\in [t_1,t_2]\)  and \(y_1\in (\gamma _1(x)-\epsilon , \gamma _1(x))\) .
Now putting together \((\ref {dise01})\) , \((\ref {dise02})\)  and \((\ref {dise03})\) , dividing by \(\epsilon ^2\)  and getting to the limit to \(\epsilon \rightarrow 0\) , we obtain
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) -\phi (t_1,\hat{x}_j , \gamma _j(t_1 )) \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O} ,\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
In the similar way, using again (REF ) and (REF ) we have
\(\phi (t_2,\hat{x}_j , \gamma _j(t_2)) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) \ge - \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1)\) 
for \(t_1,t_2 \in (-T,T)\)  with \(t_1<t_2\) .
Hence combining (REF ) and (REF ) we get
\(|\phi (t_2,\hat{x}_j , \gamma _j(t_2 )) - \phi (t_1,\hat{x}_j , \gamma _j(t_1)) | \le \biggl (\Vert w_j\Vert _{\mathcal {L}^\infty (\mathcal {O},\mathbb {R})} + \frac{(1+\sqrt{2})C_h^2}{2} \Big (|b_{j1}^{(1)} |+ |b_{j1}^{(2)}|\Big ) \biggl ) (t_2-t_1),\) 
for \(t_1,t_2 \in (-T , T)\)  with \(t_1<t_2\) , i.e. \(\phi \)  is a Lipschitz map along any characteristic line \(\gamma _{j}\)  as desired.
